Superman and Lois Lane had already had one son, Joel, who lost his powers in the womb thanks to exposure to gold kryptonite, so when they gave birth to a superpowered baby girl they had quite a problem. First they had to teach baby Kara how to use her powers, and keep them a secret from Joel so he wouldn't be jealous. As a teenager, Kara used her powers as Supergirl, disguising her blonde hair with a black wig. She formed a Justice League with the children of other heroes, and fell in love with Batman's son Bruce Jr. Upon reaching adulthood, Kara was re-christened Superwoman and engaged to Bruce Jr. who was now Batman. However nobody knew that Lex Luthor had told Joel all about his parents deceptions and filled his head with lies and gave him an experimental serum that gave the evil, twisted Joel powers of his own, powers that enabled him to kill his sister before dying due to the poisonous nature of the serum.
Kara has all of the powers of Superman (Superhuman strength, superhuman speed, flight, invulnerability, x-ray vision, heat vision, super-hearing, super-sight, and cold breath) but her power levels are at half the capacity of a normal Kryptonian due to 50% of her DNA coming her human mother.